Again, the existence part is shown in [1]} in the special case of metrics of constant curvature \(K\in (-1,0)\) . However, this result does not include the case of pleated surfaces, for which one would like to prescribe the bending measured laminations on the two connected components of the boundary of the convex hull of a quasicircle in \(\partial _\infty {\mathbb {H}}^3\) . A possible statement is suggested by [2]}, which gives an existence result for the corresponding problem in anti-de Sitter space, see Section . We will say that two measured laminations \(\lambda _-,\lambda _+\)  on the hyperbolic disk \({\mathbb {D}}\)  strongly fill if, for every \(\epsilon >0\) , there exists \(c>0\)  such that for any geodesic segment \(\gamma \)  in \({\mathbb {D}}\)  of hyperbolic length at least \(c\) , \(i(\lambda _-,\gamma )+i(\lambda _+,\gamma )\ge \epsilon \) . We also need to remember that a measured lamination on \({\mathbb {D}}\)  can be considered as a measure on \({\mathbb {RP}}^1\times {\mathbb {RP}}^1\setminus \Delta \) , where \(\Delta \subset {\mathbb {RP}}^1\times {\mathbb {RP}}^1\)  is the diagonal, and notice that given a quasicircle \(C\subset \partial _\infty {\mathbb {H}}^3\) , the measured bending laminations on the two connected components of boundary the convex hull of \(C\)  can be considered as measures on \(C\times C\setminus \Delta \) , where again \(\Delta \subset C\times C\)  is the diagonal.
